Circles
Area, circumference, and slices of a circle, in terms of π.
In your head, jot if needed · no calculator why?
Opens at level 22. You're level 1. You can read and practice here now.
the idea
Every circle measurement comes from its radius r, the distance from the centre to the edge. The diameter d goes all the way across, so it is twice the radius.
The circumference, the distance around, is 2πr, the same as πd. The area is πr². Answers here are written as a number times π, like 25π, so you never need a decimal for π.
A sector is a slice, like a slice of pizza. Its angle says what fraction of the full 360° it takes, and it gets that fraction of the area and of the circumference.
techniques
Radius first
- Given the diameter? Halve it to get the radius.
- Area: square the radius. The number times π is the area.
- Circumference: double the radius, which is the diameter. That number times π is the circumference.
worked example
A circle has diameter 14. Its area is kπ. What is k?
- Radius: 14 ÷ 2 = 7.
- Square it: 7 × 7 = 49, so the area is 49π.
Answer: 49
Take the slice's fraction
- Write the angle over 360 and simplify: 90° is 1/4, 120° is 1/3.
- Sector area: that fraction of πr².
- Arc length: that fraction of 2πr. A fraction answer is fine.
worked example
A sector of a circle has radius 6 and a central angle of 60°. Its area is kπ. What is k?
- 60/360 = 1/6, so the sector is one sixth of the circle.
- Whole circle: 6 × 6 = 36, so its area is 36π.
- One sixth of 36π is 6π.
Answer: 6
watch out for
- Using the diameter as the radius. Halve it first, or the area comes out four times too big.
- Mixing the formulas: 2πr is the distance around, and πr² is the area inside.
- Giving the arc length when the question asks for the sector's area, or the reverse.
practice
Circle area
worked example
A circle has diameter 2. Its area is kπ. What is k?
Answer: 1
- The radius is half the diameter: 1.
- Area = πr² = 1² × π = π, so k = 1.
Circumference
worked example
A circle has diameter 15. Its circumference is kπ. What is k?
Answer: 15
- Circumference = πd = 15π, so k = 15.
Sectors & arcs
worked example
A sector of a circle has radius 2 and a central angle of 45°. Its area is kπ. What is k? Give k as a whole number or a fraction.
Answer: 1/2
- 45° is 1/8 of a full turn.
- The whole circle's area is 2² × π = 4π.
- The sector is 1/8 × 4π = (1/2)π, so k = 1/2.